PaperlayerView exact revision on arXiv

Source-linked AI summary

Symplectic filtering of Hamiltonian dynamics with moving sensors

Olga Mula, Cecilia Pagliantini, Federico Vismara

arXiv:2609.35647v1math.NAcs.NAmath.DSmath.SG

TL;DR

Filtering Hamiltonian systems has less-developed mathematical foundations than forward simulation, particularly for rigorous accuracy guarantees. This paper develops an online method that updates symplectic approximation spaces using measurements and can move sensors to improve stability. Its analysis establishes error bounds and geometric preservation, while experiments show benefits from adaptive sensor placement.

  • Problem

    Filtering Hamiltonian systems remains less developed than structure-preserving forward simulation, and recovering a state from partial observations is ill-posed without additional information.

    Source: §1 · p. 1, §1.1 · p. 2

  • Method

    The online method evolves low-dimensional symplectic approximation spaces using PDE dynamics and measurement data, with sensor locations adapted to maximize the stability constant.

    Source: §6 · p. 22, §8 · p. 30

  • Results

    The reconstruction belongs to a symplectic space and preserves energy up to approximation error; shallow-water experiments show benefits of adaptive sensor placement over static observations.

    Source: §8 · p. 30

  • Takeaways & Limitations

    The analysis identifies stability as central to reconstruction accuracy and supports dynamically adapting sensor locations to improve measurement-space alignment.

    Source: §6 · p. 22, §8 · p. 30

    The framework assumes the target solves a parametric PDE with an unknown parameter; parameter estimation and nonlinear observation operators remain future directions.

    Source: §1.1 · p. 3, §8 · p. 30

Abstract

from arXiv · show

While structure-preserving methods for the forward simulation of Hamiltonian systems are well established, the mathematical foundations of their data assimilation and filtering counterparts remain comparatively less developed, particularly regarding rigorous accuracy guarantees. To address this gap, we propose an online filtering algorithm to reconstruct an unknown function $u^\dagger$ from finitely many measurements, assuming that $u^\dagger$ solves a parametric Hamiltonian PDE with unknown inputs. The method uses low-dimensional symplectic approximation spaces that evolve in time according to the PDE model and informed by the measurement data. The method can also be coupled with a dynamical sensor-placement strategy designed to optimize reconstruction stability. We derive error bounds in terms of stability and best approximation, and show that the reconstruction remains in a symplectic space and preserves energy up to the approximation error. Numerical experiments demonstrate the effectiveness of the method and the benefits of dynamic over static sensor placement.

1 Introduction

The paper develops an online filtering strategy for reconstructing Hamiltonian dynamics from partial measurements, addressing the limited theoretical foundations of filtering compared with forward simulation. It adapts approximation and observation spaces using incoming data, with stability analysis and moving sensors to support accurate, geometry-preserving reconstruction.

1.1 The filtering problem

  • Because finitely many observations leave infinitely many states compatible with the data when dim(H) exceeds the number of observations, reconstruction requires a priori information.Here, the prior is that the target solves a parametric evolution problem.Source: §1.1 · p. 2

1.2 Main strategy for the recovery

  • The strategy approximates the target’s velocity from velocity observations in time-dependent spaces, then integrates it to recover the state using observations accumulated through the current time.This velocity-based construction is intended to yield a C1 reconstructed state coherent with the exact solution’s regularity.Source: §1.2 · pp. 4–5

1.3 Contributions and outline of the paper

  • The method jointly evolves the approximation space using PDE dynamics and measurement data, while imposing geometric constraints and deriving stability and error bounds.Because the space is inferred during evolution, its recovery bounds are weaker than the idealized estimate assuming the space is known.Source: §1.3 · p. 5
  • Reconstruction stability depends on the alignment of approximation and observation spaces, motivating sensors that move to adapt the observation space over time.The near-optimality constant is interpreted through the angle between these spaces; fixed observations may lose stability for transport, sharp gradients, or localized structures.Source: §1.3 · pp. 5–6

1.4 Connections and novelties with respect to earlier works

  • Unlike Bayesian filtering work on Hamiltonian systems, the deterministic approach provides rigorous reconstruction error bounds and quantifies the role of measurement nature and placement.It extends time-dependent PBDW by adapting both spaces with filtering feedback, whereas the earlier framework evolved the approximation space solely from forward dynamics.Source: §1.4 · p. 6

2 Hamiltonian dynamics in Hilbert spaces

Hamiltonian systems in Hilbert spaces have conserved energy and symplectic flows. The paper uses evolving orthosymplectic bases and their tangent spaces to approximate the dynamics while preserving the symplectic structure.

2.1 Definition

  • A canonical Hamiltonian system is specified by a Hilbert space, a canonical symplectic form, and a differentiable Hamiltonian function; its flow preserves the Hamiltonian and the symplectic form.The paper gives the Schrödinger equation as an example of a Hamiltonian PDE.Source: §2.1 · pp. 6–7

2.2 The manifold of orthosymplectic functions

  • The reconstruction uses spaces spanned by orthosymplectic bases and approximates the target velocity in the bases’ evolving tangent spaces.The basis evolves along horizontal tangent directions to maintain the orthosymplectic constraints.Source: §2.2 · p. 8

2.3 Remarks on notation

  • Elements of H2n are treated as column vectors, with scalar multiplication and matrix application defined componentwise on their basis functions.The notation also omits parentheses when applying J and projection operators to such vectors.Source: §2.3 · p. 9

3 The filtering scheme

The filtering scheme first reconstructs from measurements in an initial symplectic space, then evolves that space with the modeled dynamics. Its accuracy depends on measurement stability and best approximation, while its velocity-based update is designed to give a well-defined evolution.

3 The filtering scheme

  • At t = 0, the algorithm solves a least-squares problem using measurements w(0) in H2n(0), an orthosymplectic approximation space; its error is bounded by β^-1 times the best-approximation error.Specifically, e(0) = ∥u†(0) − A(0)(w(0))∥H ≤ β^-1(H2n(0), Wm(0)) min over v∈H2n(0) ∥u†(0) − v∥H.Source: §3 · p. 9
  • For t > 0, the reconstruction follows a symplectic flow whose state lies in a time-evolving space and whose velocity approximates the target trajectory's velocity.The evolving space is sought to approximate trajectories across the parameter family, not only the unknown target trajectory.Source: §3 · p. 10
  • A direct state-level fit does not prescribe how to evolve the approximation space, so the method instead formulates the update at the velocity level.The initial velocity-based conditions alone do not determine the orthosymplectic basis evolution throughout the ambient space.Source: §3 · p. 10
  • The basis velocity is restricted to the horizontal tangent component and regularized to ensure a unique minimizer for the space-evolution problem.The regularization parameter is λ > 0.Source: §3 · p. 10

4 Reformulation with symplectic decoders

This section reformulates the filtering approximation using a symplectic decoder and derives explicit parameter dynamics for fitting target and modeled trajectories. It addresses conditions for injectivity, unique evolution, and preservation of the orthosymplectic basis, and clarifies the observation-dependent correction’s dependence on the penalization parameter λ.

4 Reformulation with symplectic decoders

  • The decoder’s two outputs represent the reconstruction and the parametrized Hamiltonian trajectories, and the filtering scheme evolves its decoder parameter over time.This reformulates the approximation strategy from Section 3 in terms of a decoder mapping.Source: §4 · p. 11

4.1 Properties of the decoder

  • The decoder is differentiable and its differential satisfies an adjoint-related inner-product identity involving the map T.The identity holds for admissible tangent perturbations and pairs in H × L.Source: §4.1 · p. 12
  • Under the stated restrictions on γ and the admissible tangent space, the differential of the decoder is injective.The result applies on the restricted tangent space when γ belongs to the specified set eΓ and S(c) is full rank.Source: §4.1 · pp. 13–14

4.2 Back to the filtering scheme: derivation of a parameter curve

  • When β(H2n(t), Wm(t)) > 0, the parameter minimization has a unique solution that defines the evolution equations for the decoder parameters.The equations govern the time-varying coefficients and basis parameters.Source: §4.2 · pp. 15–16
  • If the initial basis is orthosymplectic, the parameter evolution preserves its orthosymplecticity; the initial basis can be constructed using a symplecticity-preserving SVD algorithm.The experiments construct the initial basis from sampled PDE initial conditions.Source: §4.2 · p. 18

4.3 The case of a symplectic observation space

  • If the observation space W2m is symplectic and β(H2n(t), W2m(t)) > 0, the minimizer is given by an evolution system explicit in the parameter derivatives.This removes the implicit dependence on the derivatives of a and V present in Theorem 1’s equations.Source: §4.3 · p. 19
  • The system is equivalent to Theorem 1, but its equations are explicit in the variables’ time derivatives and highlight the role of W2m ∩ H⊥.Source: §4.3 · p. 19
  • As λ approaches infinity, (39c) becomes the observation-independent reduced-basis evolution from the forward problem [22], whereas finite λ > 0 adds a W2m correction dependent on velocity measurements.Source: §4.3 · p. 19
  • Although the target velocity appears in (39a) and (39c), evaluating their right-hand sides requires only its projections onto W2m and W2m ∩ H⊥, computable from velocity measurements.Source: §4.3 · p. 19

5 Error and stability analysis

The section derives a reconstruction error bound through an intermediate estimate and a Grönwall argument. The derivation assumes Lipschitz continuity of the Hamiltonian vector fields and analyzes the parameter-velocity minimization.

5 Error and stability analysis

  • The minimization proof decomposes trajectory velocities into components within and orthogonal to the reduced space to obtain the update terms.The derivation also minimizes over parameter velocities for fixed basis velocity.Source: §5 · pp. 20–21
  • The error analysis uses affine spaces associated with the reduced approximation to bound terms in the intermediate estimate.The affine spaces are those introduced in Lemma 6.Source: §5 · p. 21
  • A Grönwall estimate yields the reconstruction error bound after an intermediate result is established.The intermediate result assumes each Hamiltonian vector field operator Bθ is Lipschitz continuous.Source: §5 · pp. 19, 21

6 Dynamical sensor placement

The sensor-placement strategy evolves measurement locations to maximize the stability constant as the reduced space changes. This addresses the possibility that stability can deteriorate over time with a moving reduced space.

6 Dynamical sensor placement

  • The sensor trajectory is chosen to maximize the stability constant β as the reduced space evolves, because β can approach zero over time for a moving reduced space.The practical update maximizes β for the newly updated reduced space, using gradient descent initialized at the previous sensor locations.Source: §6 · p. 22
  • Sensor locations determine their measurement representers, and each sensor measures both components of the target while the resulting observation space remains symplectic.Local averages of width σ around each sensor provide the guiding example.Source: §6 · p. 22

7 Numerical experiments

The shallow-water experiments compare static random, static uniform, and dynamically moved sensors using reconstruction accuracy, stability, and Hamiltonian conservation. Dynamic sensors yield reconstruction error close to the best approximation and substantially improve stability and Hamiltonian conservation, while both static arrangements have notable shortcomings.

Experimental setup

  • The experiments use two-dimensional shallow-water equations, with n = 6, m = 8, λ = 0.1, and 100 sampled parameters; they compare random static, uniform static, and moving sensors.The grid has 5000 spatial points, and the simulation runs to T = 7 using 3500 time steps.Source: §7 · p. 23

Static sensors, random placement

  • With random static sensors, reconstruction error substantially exceeds the best approximation error, and β is around 10^-5 at the final time.The reconstructed solution is qualitatively different from the target and fails to capture the height amplitude; Hamiltonian preservation is also negatively affected.Source: §7 · p. 24, Figure 1 · p. 24, Figure 2 · p. 25

Static sensors, uniform placement

  • With uniform static sensors, maximum pointwise error is around 10%, yet both reconstructed profiles remain completely different from the target.The relative reconstruction error remains above 1% in the final stages, β is below 10^-3 for most of the interval, and Hamiltonian conservation error can reach 1.Source: §7 · pp. 25–26, Figure 4 · p. 26, Figure 5 · p. 27

Dynamic sensors

  • Around 10^-4 final reconstruction error with dynamic sensors is almost two orders of magnitude below the static-sensor case, while β stays above 10^-2 until the final time.The dynamic-sensor reconstruction error stays close to the best approximation error at all times.Source: §7 · p. 28, Figure 8 · p. 28

8 Conclusions

The paper presents an adaptive framework for reconstructing time-dependent Hamiltonian solutions from finite linear measurements, with rigorous analysis linking accuracy to stability. The reconstruction remains in a symplectic space and preserves energy up to approximation error; shallow-water experiments illustrate the benefits of adaptive sensor placement.

8 Conclusions

  • The framework incorporates measurement data into the online evolution of approximation spaces, and its error analysis identifies the stability constant β as central to reconstruction accuracy.This analysis motivates a dynamical strategy for adapting sensor locations.Source: §8 · pp. 29–30
  • The reconstruction belongs to a symplectic space and preserves energy up to the approximation error.These properties incorporate intrinsic physical features of the Hamiltonian dynamics.Source: §8 · p. 30
  • Experiments on the two-dimensional shallow water equations demonstrate practical effectiveness and benefits of adaptive sensor placement over static observations.Source: §8 · p. 30
  • Future work includes parameter estimation to evolve the probability distribution over parameters and extending the method to nonlinear measurement operators.Source: §8 · p. 30

A Proof of Theorem 2

The proof establishes identities for projections onto intersections of approximation and observation spaces, then uses symplecticity to derive the reformulated evolution equations. It shows how the reformulation yields an expression for the evolving basis.

A Proof of Theorem 2

  • When V ∩ W⊤ is trivial, the projection onto W ∩ V⊥ is characterized using the invertible matrix G(V, PWV).The proof identifies this projection through a unique projected element.Source: A Proof of Theorem 2 · p. 32
  • For a symplectic approximation space, the operator J commutes with its orthogonal projection.Source: A Proof of Theorem 2 · p. 32
  • The proof derives system (39) from system (25) when the observation space W2m is symplectic.Source: A Proof of Theorem 2 · p. 33
  • The derivation isolates terms involving V̇ and projects the resulting equation onto W2m ∩ H2n⊥ and its orthogonal complement to obtain an expression for V̇.Source: A Proof of Theorem 2 · pp. 33–35

B Spatial and temporal discretization

The numerical formulation discretizes the evolving symplectic basis, parameter samples, and sensor observations, then advances the resulting variables in time. Its time integrator preserves orthosymplecticity of the basis and the Hamiltonian structure of the coefficient evolution.

B Spatial and temporal discretization

  • The discrete basis is represented by matrices satisfying orthosymplecticity conditions, while the coefficient matrix collects basis coefficients at sampled parameters.Source: B Spatial and temporal discretization · pp. 35–36
  • The full-rank condition requires at least p ≥ 2n parameter samples; the work assumes this condition holds.Without it, the evolution for the coefficient matrix may be rank-deficient.Source: B Spatial and temporal discretization · p. 36
  • The numerical experiments evaluate the observation-space basis from sensor locations when the Riesz representers are known; otherwise, they would need to be computed by solving a boundary-value problem.Source: B Spatial and temporal discretization · p. 36
  • Measurements and sensor locations are held fixed within each time step, and a second-order partitioned Runge–Kutta scheme advances the discrete variables.Implicit midpoint for Z preserves the coefficient evolution's Hamiltonian structure, while the method keeps U orthosymplectic at time steps.Source: B Spatial and temporal discretization · p. 36
  • The initial basis is constructed by complex SVD of Y0, then Z0 and the initial coefficient vector a0 are computed from the initial data and measurements.Source: B Spatial and temporal discretization · p. 37
Loading 2609.35647v1…